Regarding physics computations:

To make the experience smoother we will use a testing framwork called check

Newton II solver, was created, to compute a projectile's trajectory under gravity and air resistance $($drag$)$ using initial position and velocity. Each time step updates position and velocity based on acceleration from gravity and drag:

where $\backslash$alpha is proportional to drag and $\backslash$beta to particle mass.

Our solver is currently broken. Your task: find and fix the bug so all tests pass. Use unit tests as guidance. See the section "The makefile and project structure" for running instructions. If the program hangs, press CTRL$+$C to stop.

$$

$$

Main file:

#include $//$ printf

#include $//$ malloc

#include "vector.h$"$

$$

void

print$\_$vector$($

$$ double $*$vec$,//$ Vector to print

$$ unsigned int ndims$//$ Number of dimensions

$)$

$\{$

$$ for $($int i $=0$; i $<$ ndims; i$++)\{$

$$ printf$("\%10\mathrm{.}5$f $",$ vec$[$i$])$;

$\}$

$\}$

void

solve$\_$general$($

$$ double $*$pos$,//$ Position vector. Initialize with initial position

$$ double $*$vel$,//$ Velocity vector. Initialize with initial velocity

$$ double $*$gravity$,//$ Vector with the opposite direction of gravity

$$ unsigned int ndims$//$ Number of dimensions of vectors

$)$

$\{$

$//$ Constants

$$ const double dt $=0\mathrm{.}01$f;$//$ Simulation time step

$$ const double alpha $=1\mathrm{.}0$f;$//$ Proportional to drag coefficient

$$ const double beta $=1\mathrm{.}0$f;$//$ Proportional to mass

$$

$$ double $*$tmp $=($double $*)$malloc$($sizeof$($double$)*$ndims$)$;$//$ Buffer for storing intermediate work

$$ double time $=0\mathrm{.}0$f;$//$ Simulation time

$$

$//$ Print header

$$printf$("$# Column $0$: Time

$")$;

$$ printf$("$# Columns $\%$i$-\%$i: Position

$",1,$ ndims$)$;

$$ printf$("$# Columns $\%$i$-\%$i: Velocity

$",$ ndims $+1,$ ndims $*2)$;

$$

$//$ Print positions and velocity

$$ print$\_$vector$($&time, $1)$;

$$ print$\_$vector$($pos$,$ ndims$)$;

$$ print$\_$vector$($vel$,$ ndims$)$;

$$ printf$("$

$")$;

$$

$$ while$(1)\{$

$//$ Update position by adding velocity times dt

$$ constant$\_$multiplication$($tmp$,$ vel, dt$,$ ndims$)$;$//$ tmp $<-$ v$*$dt

$$ elementwise$\_$addition$($pos$,$ pos, tmp$,$ ndims$)$;$//$ pos $<-$ pos $+$ v$*$dt

$$

$//$ Compute acceleration from drag, multiply with dt and add to velocity

$$ constant$\_$multiplication$($tmp$,$ vel, $-$alpha, ndims$)$;

$$ constant$\_$multiplication$($tmp$,$ tmp$,$ dt$,$ ndims$)$;

$$ elementwise$\_$addition$($vel$,$ vel, tmp$,$ ndims$)$;

$$

$//$ Compute acceleration from gravity, multiply with dt and add to velocity

$$ constant$\_$multiplication$($tmp$,$ gravity, $-$beta, ndims$)$;

$$ constant$\_$multiplication$($tmp$,$ tmp$,$ dt$,$ ndims$)$;

$$ elementwise$\_$addition$($vel$,$ vel, tmp$,$ ndims$)$;

$$

$//$ Update time

$$ time $+=$ dt;

$$

$//$ Print positions and velocity

$$ print$\_$vector$($&time, $1)$;

$$ print$\_$vector$($pos$,$ ndims$)$;

$$ print$\_$vector$($vel$,$ ndims$)$;

$$ printf$("$

$")$;

$$

$//$ Break the simulation when below $0$ in the dimension of gravity

$$ if $($dot$\_$product$($pos$,$ gravity, ndims$)<0)\{$

$$ break;

$\}$

$\}$

$$

$$ free$($tmp$)$;

$$ tmp $=$ NULL;

$\}$

$$

void

solve$\_3$d$()\{$

$$ double pos$[]=\{0.$f$,0.$f$,0.$f$\}$;$//$ Position

$$ double vel$[]=\{1.$f$,1.$f$,0\mathrm{.}3$f$\}$;$//$ Velocity

$$ double gravity$[]=\{0.$f$,1.$f$,0.$f$\}$;$//$ Gravity is directed in negative y

$$

$$ solve$\_$general$($pos$,$ vel, gravity, $3)$;

$\}$

$$

int

main$()$

$\{$

$$ solve$\_3$d$()$;

$$ return $0$;

$\}$

$------------------$

vector file

#include

$$

#include "vector.h$"$

$$

void

elementwise$\_$addition$($

$$ double $*$res$,$

$$ double $*$v$1,$

$$ double $*$v$2,$

$$ unsigned int len

$)$

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